A) 0
B) -1
C) 1
D) 2
Correct Answer: D
Solution :
\[\left| \begin{matrix} a & b-y & c-z \\ a-x & b & c-z \\ a-x & b-y & c \\ \end{matrix} \right|=0\] \[\Rightarrow \]\[a\left| \begin{matrix} b & c-z \\ b-y & c \\ \end{matrix} \right|-(b-y)\left| \begin{matrix} a-x & c-z \\ a-x & c \\ \end{matrix} \right|\] \[+(c-z)\left| \begin{matrix} a-x & b \\ a-x & b-y \\ \end{matrix} \right|=0\] \[\Rightarrow \]\[a(bc-bc+bz+cy-yz)-(b-y)\] \[\left( ac-cx-ac+az+cx-xz \right)+\left( c-z \right)\] \[\left( ab-ay-bx+xy-ab+bx \right)=0\] \[\Rightarrow \]\[a(bz+cy-yz)-(b-y)(az-zx)+(c-z)\] \[(xy-ay)=0\] \[\Rightarrow \]\[abz+acy-ayz-abz+bxz+ayz-xyz\] \[+cxy-acy-xyz+ayz=0\] \[\Rightarrow \]\[ayz+bxz-2xyz+cxy=0\] \[\Rightarrow \]\[ayz+bxz+cxy=2xyz\] \[\Rightarrow \]\[\frac{ayz}{xyz}+\frac{bxz}{xyz}+\frac{cxy}{xyz}=2\] \[\Rightarrow \]\[\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=2\]
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